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Unformatted text preview: DIRICHLET’S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS PETE L. CLARK 1. Statement of Dirichlet’s theorem The aim of this section is to give a complete proof of the following result: Theorem 1. (Dirichlet, 1837) Let a,N ∈ Z + be such that gcd( a,N ) = 1 . Then there are infinitely many prime numbers p such that p ≡ a (mod N ) . We remark that the proof gives more, that the set of primes p ≡ a (mod N ) is substantial in the sense of [Handout 12]. 1 One of the amazing things about the proof of Dirichlet’s theorem is how modern it feels. It is literally amazing to compare the scope of the proof to the arguments we used to prove some of the other theorems in the course, which historically came much later. Dirichlet’s theorem comes 60 years before Minkowksi’s work on the geometry of numbers and 99 years before the Chevalley-Warning theorem! Let us be honest that the proof of Dirichlet’s theorem is of a difficulty beyond that of anything else we have attempted in this course. On the algebraic side, it requires the theory of characters on the finite abelian groups U ( N ) = ( Z /N Z ) × . From the perspective of the 21st century mathematics undergraduate with a back- ground in abstract algebra, these are not particularly deep waters. More serious demands come from the analytic side: the main strategy is, as in Euler’s proof of the infinitude of primes, to consider the function P a ( s ) = X p ≡ a (mod N ) 1 p s , which is defined say for real numbers s > 1, and to show that lim s → 1 + P a ( s ) = + ∞ . Of course this suffices, because a divergent series must have infinitely many terms! The function P a ( s ) will in turn be related to a finite linear combination of logarithms of Dirichlet L-series, and the differing behavior of the Dirichlet series for principal and non-principal characters is a key aspect of the proof. Indeed, the fuel for the entire proof is the following surprisingly deep fact: Theorem 2. (Dirichlet’s Nonvanishing Theorem) For any non-principal Dirichlet character χ of period N , we have L ( χ, 1) 6 = 0 . There are many possible routes to Theorem 2. We have chosen (following Serre) to present a proof which exploits the theory of Dirichlet series which we have de- veloped in the previous handout in loving detail. As in our treatment of Dirichlet 1 In fact, with relatively little additional work, one can show that the primes are, in a certain precise sense, equidistributed among the ϕ ( N ) possible congruence classes. 1 2 PETE L. CLARK series, we do find it convenient to draw upon a small amount of complex function theory. These result are summarized in Appendix C, which may be most useful for a reader who has not yet been exposed to complex analysis but has a good command of the theory of sequences and series of real functions....
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This note was uploaded on 10/26/2011 for the course MATH 4400 taught by Professor Staff during the Spring '11 term at UGA.
- Spring '11
- Number Theory